A strong topological phase does not rely on the microscopic translation symmetry for its topological protection.
In particular, it is independent of the size of the unit cell.In topological band theory, such as developed in Ref. [1]}, strong topological phases can be described in the long-wavelength expansion of the single-particle Hamiltonian. To connect to this approach, one needs to require that the momentum is well-defined in the long wavelength limit. As only the long-wavelength limit is required, the size of the unit cell can be chosen arbitrarily large.
This allows to coarse-grain the lattice and switch to an effective, continuous description (c.f. also Ref. [2]} for a detailed discussion).
For strong topological phases, we only need to require that the topological properties are realized when the system size is much larger than any microscopic length scale associated with the Hamiltonian.
Consequently, during a Volterra process of a topological crystalline phase, we can assume that also the cut-out part is in the same topological phase.
